source file: mills2.txt Date: Sat, 22 Mar 1997 20:05:13 -0800 Subject: Meantones and ETs - correction From: kollos@cavehill.dnet.co.uk (Jonathan Walker) Although I said that I sent the "Pi;meantones and ETs" message to the list accidentally, I do feel I ought to warn anyone who bothered to note down the formula that there was an error, namely a missing bracket. My thanks to Jonathan Wild for noticing that there must have been a typo in Friday's version. What I had previously sent was: > |((nlog_2(3^((4-m)/m)5^(-1/m))-4/m))/[nlog_2(3)])+1| < 0.00011 This should have read: |((n(log_2(3^((4-m)/m)5^(-1/m))-4/m))/[nlog_2(3)])+1| < 0.00011 So there is an extra bracket between the first "n" and "log_2". For greater legibility, I'll provide the following version. | (4-m)/m -1/m | | n(log_2(3 * 5 )) - 4/m) | | ____________________________________ + 1 | < 0.00011 | | | [nlog_2(3)] | Just to remind anyone who might be interested, m is the root of the syntonic comma in the meantone system for which an ET approximation is being sought, while n is the number of steps per octave in the ET being tested; [x] is the integer value of x. Here are the margin-of-error figures I've obtained for the six historic meantones and their standard ET near-equivalents, in the form (1/m,n): (1/3,19) 0.0000260699 (2/7,50) 0.0001000280 (1/4,31) 0.0001032090 (2/9,74) 0.0000640236 (1/5,43) 0.0000108953 (1/6,55) 0.0000994642 The margin of error which I have given (<0.00011) can, of course, be altered in either dirction, depending on the level of accuracy you want; I've set it so that it easily catches all of the above standard near-equivalents. -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 23 Mar 1997 22:14 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA12125; Sun, 23 Mar 1997 22:14:10 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA12120 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id NAA10894; Sun, 23 Mar 1997 13:11:39 -0800 Date: Sun, 23 Mar 1997 13:11:39 -0800 Message-Id: <199703232109.NAA10676@ella.mills.edu> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu